CSE115/ENGR160 Discrete Mathematics02/28/12

Ming-Hsuan Yang

UC Merced

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Insertion sort

• Start with 2nd term– Larger than 1st term, insert after 1st term– Smaller than 1st term, insert before 1st term

• At this moment, first 2 terms in the list are in correct positions• For 3rd term

– Compare with all the elements in the list– Find the first element in the list that is not less than this element

• For j-th term– Compare with the elements in the list– Find the first element in the list that is not less than this

element

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Example

• Apply insertion sort to 3, 2, 4, 1, 5• First compare 3 and 2 2, 3, 4, 1, 5• Next, insert 3rd item, 4>2, 4>3 2, 3, 4, 1, 5• Next, insert 4th item, 1<2 1, 2, 3, 4, 5• Next, insert 5th item, 5>1, 5>2, 5>3, 5>41, 2,

3, 4, 5

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Insertion sortprocedure insertion sort(a1, a2, …, an: real numbers with n≥2)

i:=1 (left endpoint of search interval) j:=1 (right end point of search interval) for j:=2 to n begin i:=1 while aj>ai

i:=i+1 m:=aj

for k:=0 to j-i-1

aj-k:= aj-k-1

ai:= m

end {a1 , a2, …, an are sorted}

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Greedy algorithm

• Many algorithms are designed to solve optimization problems

• Greedy algorithm: – Simple and naïve– Select the best choice at each step, instead of

considering all sequences of steps– Once find a feasible solution– Either prove the solution is optimal or show a

counterexample that the solution is non-optimal

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Example

• Given n cents change with quarters, dimes, nickels and pennies, and use the least total number of coins

• Say, 67 cents• Greedy algorithm

– First select a quarter (leaving 42 cents)– Second select a quarter (leaving 17 cents)– Select a dime (leaving 7 cents)– Select a nickel (leaving 2cnts)– Select a penny (leaving 1 cent)– Select a penny

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Greedy change-making algorithm

procedure change(c1, c2, …, cn: values of denominations of coins, where c1>c2>…>cn; n: positive integer)

for i:=1 to r while n≥ci then

add a coin with value ci to the change

n:=n- ci

end7

Example

• Change of 30 cents• If we use only quarters, dimes, and pennies

(no nickels)• Using greedy algorithm:

– 6 coins: 1 quarter, 5 pennies – Could use only 3 coins (3 dimes)

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Lemma 1

• If n is a positive integer, then n cents in change using quarters, dimes, nickels, and pennies using the fewest coins possible has at most two dimes, at most one nickel, at most 4 pennies, and cannot have two dimes and a nickel

• The amount of change in dimes, nickels, and pennies cannot exceed 24 cents

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Proof (Lemma)

• Proof by contradiction• Show that if we had more than the specified number of coins

of each type, we could replace them using fewer coins that have the same value– If we had 3 dimes, could replace with 1 quarter and 1 nickel– If we had 2 nickels, could replace them with 1 dime– If we had 5 pennies, could replace them with 1 nickel– If we had 5 pennies, could replace them with 1 nickel– If we had 2 dimes and 1 nickel, could replace them with 1 quarter– Because we could have at most 2 dimes, 1 nickel, and 4 pennies, but

we cannot have two dimes and a nickel, it follows 24 cents is the most we can have

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Theorem

• Theorem: The greedy change-making algorithm produces change using the fewest coins possible

• Proof by contradiction• Suppose that there is a positive integer n such that

there is a way to make change for n cents using fewer coins (q’) than that of the greedy algorithm

• Let the number of quarters be q’, and the number of quarters used in the greedy algorithm be q

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Proof

• First note q’ must be the same as q• Note the greedy algorithm uses the most

quarters possible, so q’≤q• However, q’ ≮ q• If q’ < q, we would need to make up 25 cents from dimes, nickels, and pennies in the optimal way to make change• But this is impossible from Lemma 1

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Proof

• As there must be the same number of quarters in the two algorithms

• The value of the dimes, nickels and pennies in these two algorithms must be the same, and their value is no more than 24 cents

• Likewise, there must be the same number of dimes, – as the greedy algorithm used the most dimes possible – and by Lemma 1, when change is made using the fewest coins

possible, at most 1 nickel and a most 4 pennies are used, so that the most dimes possible are also used in the optimal way to make change

• Likewise, we have the same number of nickels, and finally the same number of pennies

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The halting problem

• One of the most famous theorems in computer science

• There is a problem that cannot be solved using any procedure

• That is, we will show there are unsolvable problems

• The problem is the halting problem

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The halting problem

• It asks whether there is a procedure that does this: – It takes input as a computer program and input to

the program, and – determines whether the program will eventually

stop when run with the input

• Useful to test certain things such as whether a program entered into an infinite loop

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The halting problem

• First note that we cannot simply run a program and observe what it does to determine whether it terminates when run with the given input

• If the program halts, we have our answer• But if it is still running after any fixed length of

time has elapsed, we do not know whether it will never halt or we just did not wait long enough for it to terminate

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Turing’s proof

• Halting problem is unsolvable• Proof by contradiction• The proof presented here is not completely rigorous• Proof: Assume there is a solution to this halting

problem called H(P,I) where P is a program and I is input

• H(P,I) outputs the string “halt” as output if H determines P stops when given I

• Otherwise, H(P,I) generates the string “loops forever” as output

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Turing’s proof

• When a procedure is coded, it is expressed as a string of characters and can be interpreted as a sequence of bits

• A program can be used as data, and thus a program can be thought of as input to another program, or even itself

• H can take a program P as both of its inputs, which are a program and input to this program

• H should be able to determine if P will halt when it is given a copy of itself as input

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Turing’s proof

• Construct a simple procedure K(P) that makes use of the output H(P,P) but does the opposite of H

• If the output of H(P,P) is “loops forever”, then K(P) halts• IF the output of H(P,P) is “halt”, then K(P) loops forever

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Turing’s proof

• Suppose we provide K as input to K• We note that if the output of H(K,K) is

“loops forever”, then by the definition of K, we see K(K) halts

• Otherwise, if the output of H(K,K) is “halt”, then by the definition of K we see that K(K) loops, in violation of what H tells us

• In both cases, we have contradiction• Thus H cannot always give the correct answers• No procedure solves the halting problem

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